Advances in High Energy Physics
Volume 2022 (2022), Article ID 6949022, 11 pages
https://doi.org/10.1155/2022/6949022

Research Article

The Nonsymmetric Flavor Transition Matrix and the Apparent P Violation

Shu-Jun Rong and Ding-Hui Xu

College of Science, Guilin University of Technology, Guilin, Guangxi 541004, China

Correspondence should be addressed to Shu-Jun Rong; rongshj@glut.edu.cn

Received 16 December 2021; Accepted 25 February 2022; Published 12 March 2022

Academic Editor: Sunny Vagnozzi

Copyright © 2022 Shu-Jun Rong and Ding-Hui Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

Abstract.

The leptonic mixing parameters of high precision and the next-generation neutrino telescopes make it possible to test new physics in the flavor transition of the high-energy astrophysical neutrinos (HAN). We introduce a nonsymmetric matrix to modify the predictions of the standard flavor transition matrix. It is constructed with the mixing matrix in vacuum and that at the source of the HAN. The mismatch of the mixing matrices results in the new expectation of the flavor ratio of the HAN at Earth. It also leads to a secondary effect called the apparent P violation (APV). The quantitative analyses of the new effects are performed with a moderate setup of the parameters at the source of the HAN. The correlations between the mixing parameters and the new predictions are shown. From the correlations, the dominant parameters determining the new-physics effects are identified.

1. Introduction

Since the late 1990s, notable progresses have been made in the measurement of neutrino oscillation parameters [1–3]. According to the global analysis NuFIT 5.0 with the Super-Kamiokande atmospheric neutrino data, the relative precision of leptonic mixing angles and Dirac CP phase is shown as follows [4]:

Obviously, we have the well-measured and the uncertain . In the following decades, the precision of the mixing parameters will be improved by results from the oscillation experiments such as JUNO [5], Hyper-Kamiokande (HK) [6], DUNE [7], etc. Then, new physics on neutrinos may be tested. As we know, new physics usually lies in the extreme environment or the extreme parameter range. Recently, a large number of high-energy astrophysical neutrinos (HAN) events in the energy range of Tev-Pev have been reported by the IceCube Collaboration [8–11]. These events may provide information on new physics.

In this paper, we are concerned about the effects of new physics in the flavor transition of the HAN. Because the HAN detected at Earth have traveled cosmological-scale distances, the decoherence effects of flavor neutrinos should be considered. So the flavor conversion probability is of the form

where . is the element of the leptonic mixing matrix [12]. The so-called standard flavor transition matrix is symmetric, which is obtained on the base of the assumption that the mixing matrix of the HAN at their source and that in their journey are the same as that in vacuum, namely, . Any scenario beyond the assumption expressed by a different flavor transition matrix could manifest a class of new physics. Typical new physics includes the energy-dependent leptonic mixing matrix [13, 14], neutrino-decay [15–22], nonunitary mixing [23–30], pseudo-Dirac neutrinos [31–36], neutrino secret interactions [37–40], etc. [41]. These new-physics-based transition matrices predict various of flavor ratios of the HAN at Earth. However, correlations between the parameters in the transition matrix and the new predictions are usually unclear in these scenarios. In this paper, we propose a simplified nonsymmetric flavor transition matrix and analyse the relationship between the new effects and the mixing parameters.

In principle, matter interactions [42–45], nonstandard interactions [46–48], and other new physics may modify the leptonic mixing matrix at the source of the HAN. Similar to the case of the solar neutrinos [49], we suppose that the HAN in their motherland undergo an adiabatic flavor evolution. Then, we can generalize the flavor transition probability as

Here, is the leptonic mixing matrix at the location where the HAN are produced. Although is not of the most general form, it could describe several classes of nonstandard flavor conversion scenarios in the adiabatic cases. As an example, when is unitary, it could denote a mixing matrix including the effect of matter interaction or nonstandard interactions. In this paper, we focus on the model-independent properties and predictions of .

In general, the flavor transition matrix is nonsymmetric. It may give different predictions on the HAN flavor ratio at Earth. Furthermore, suppose that we interchange the departure location and the arrival location of neutrinos. Namely, we make a P transformation , with , 2, 3. The transition probability of the -flavor neutrinos departing from Earth would be expressed as . In the case , we have . So an apparent P violation (APV) quantified by would be observed. The APV means that the P violation is from the environment where neutrinos are produced. Furthermore, we note that . So the APV is equal to the apparent T violation (ATV).

In practice, the new effects from are impacted by the uncertainties of the parameters in . We have no precision information on which is dependent on the specific source of the HAN. So we employ a model-independent method to describe . In the standard parametrization [12], is dependent on the mixing angles () and the CP phase . We consider the case that the relative deviations of these parameters from their counterparts in vacuum are moderate, i.e., of the order of . These setups of enable us to find out the correlation between the new predictions and the leptonic mixing parameters. These observations may hold for neutrinos produced in the realistic astrophysical sources such as galactic halos and cluster halos where the matter effect on is modest [50].

The paper is organised as follows. In Section 2, the properties of the generalized flavor transition matrix are studied. On the basis of these properties, a generalized area theorem of the flavor triangle at Earth is derived. It reveals how the flavor ratio distribution at Earth is dependent on the determinant of . The expression of the APV is given in this section. The possible case that the APV may be detected is discussed. In Section 3, we perform numerical analyses of the impacts of . The correlation between the determinant of the matrices and that of is given. The quantitative flavor ratio distribution at Earth on the basis of and the special initial flavor ratios are obtained. The correlations between the APV and the mixing parameters are analysed. Finally, we conclude.

2. The Generalized Flavor Transition Matrix and New-Physics Effects

2.1. Flavor Ratio Vector and Flavor Triangle

The general neutrino flavor ratio could be expressed with a column vector, namely,

which satisfies the following conditions

So the set of the points of all the flavor ratio vectors forms a triangle in the 3-dimensional Euclidean space. Following the convention of Ref. [51], we call it flavor triangle, see Figure 1. The vertexes of the flavor triangle correspond to the vectors of the form

Figure 1: The schematic figure of the flavor triangle and the Earth flavor triangle (EFT).

The area of the flavor triangle is . Note that and correspond to the flavor ratio at the neutron-decay source and that at the muon-damping source, respectively.

Suppose that the flavor ratio of the HAN at the source is given by . When the HAN arrive at Earth, is changed to . It is connected to through the flavor transition matrix, namely,

Here, the expression of is shown in Eq. (3). After the flavor transition of HAN, the flavor triangle is transformed to that at Earth. We call it Earth flavor triangle (EFT). The schematic figure of the EFT is also shown in Figure 1.

Given the matrix , the EFT shows the general flavor ratio distribution of the HAN at Earth. The vertexes of the EFT correspond to the vectors , , and . They are the first, the second, and the third column vector of the flavor transition matrix , respectively, namely,

The area of the EFT is determined by the determinant of . The specific expression is shown in the following part.

2.2. Unitarity and the Generalized Area Theorem for the EFT

In this paper, we consider the case that the mixing matrices and are unitary. So the columns vectors , , and satisfy the constraints

where is the democratic vector of the form

On the basis of unitarity conditions Eqs. (9) and (10), we can derive an identity for the column vectors of , namely,

Here, Det denotes the determinant, . The dot product and cross product of flavor vectors are defined in the standard way, namely,

where is the totally antisymmetric tensor with . The proof of the identity is given as follows.

In the case that the column vectors are linearly independent, namely, , we can decompose the circular cross product of the vectors as

Then employing the identity

we obtain the following equations

According to the constraints Eqs. (9) and (10), we find that is the eigenvector of the coefficient matrix with the eigenvalue equal to 1. Because of the condition , we obtain the result . So the identity is proved.

Now we consider the case that column vectors are linearly dependent, namely, . On the basis of the constraints Eqs. (9) and (10), vectors and could be decomposed as

Substituting these expressions into the circular cross product, we obtain the following result:

So the identity also holds in this case.

Employing the identity, we can obtain the area of the EFT. The area vector of the EFT is expressed as

So the area of the EFT is derived, namely,

This formula is called the area theorem in Ref. [51] in the case that the transition matrix is of the form Eq. (2). Our derivations reveal that the theorem results from the constraints Eqs. (9) and (10). Furthermore, we can see that if two matrices and meet the constraints, so does their product . Therefore, besides of and , this theorem holds for a general class of flavor transition matrices. So we call it the generalized area theorem for the EFT. It is useful for the analysis of the uncertainties of the flavor ratio at the source of the HAN.

2.3. The Determinant of

The generalized theorem demonstrates that the area of the EFT is determined by the determinant of . In order to analyse the impacts of the matrix on , we show the expression of . Following the standard parametrization of the leptonic mixing matrix [12], is written as

where , . Here, the Majorana phases are not shown because they are irrelevant to the flavor transition matrix. Similarly, the mixing matrix is expressed as

where , . The superscript labels the parameters at the source of the HAN. According to the expression of , namely, Eq. (3), is of the form

where

2.4. The Apparent P Violation from

A notable characteristic of the transition matrix is the nonsymmetry. The APV could be measured by , , or . According to the unitary conditions, namely, Eqs. (9) and (10), we find the following identities

So we choose to describe the APV. In the standard parametrization, is expressed as

where are quantities dependent on the mixing angles in the matrices and . The relationship between and the mixing parameters is obtained with the numerical analysis in the next section.

In order to detect the APV, we should find out the relationship between and the flavor ratio vector at Earth. Note that is dependent not only on but also on the specific source of the HAN. Here, we consider the case that the HAN at Earth are from two different sources, namely, the pion-decay source and the muon-damping source , where is of the form

is just the vector shown in Eq. (6). We suppose that the neutrino telescopes in the future could discriminate the sources and with the help of other astronomical observatories. Furthermore, we assume that the difference between the flavor transition matrix of the source and that of is negligible, namely, ==. Under this condition, the flavor ratio at Earth with the HAN from and that from are expressed, respectively, as

where and are the column vectors of the matrix shown in Eq. (8). Then, the relationship between the APV and the flavor ratios at Earth is obtained, namely,

In general cases, the assumption = may not hold. The APV cannot be identified through the measurement of and . Even so, the term at the right hand of Eq. (31) could still serve as an index of the deviation from . Namely, if this term is nonzero, we can determine that the flavor transition matrix of the HAN is not .

3. Numerical Analyses of Impacts of

In this section, we perform numerical analyses to study the impacts of the mixing parameters in on the neutrino flavor ratio at Earth and the APV. Because of the lack of the information of the source of the HAN, we adopt a model-independent method to describe . In order to identify the correlation between the parameters in and the new predictions of , we consider the case that the relative deviation of from the vacuum mixing matrix is moderate, namely, of the order of . The specific setup of the parameters of is expressed as follows:

The parameters and are taken from the global fit data at level [4]. The modifications of the mixing parameters are randomly taken in the given ranges.

3.1. Uncertainties of the Flavor Ratios at the Source with the Compression Map

We scanned the determinants of and with the mixing parameters given in the above setup. For the comparison of the impacts of the different parameters, we modify one parameter at a time. The correlation of and is shown in Figure 2.

Figure 2: The correlation between and . The mixing angles and the CP phase in are in the ranges of the global fit data of the normal mass ordering at level [4]. The results are similar in the case of the inverted mass ordering. The modified parameter in is randomly taken in the corresponding range in Eqs. (32)–(35).

We can see that the range of the magnitude of is approximate to that of . According to the generalized area theorem for the EFT, the ratio is in the range . Considering the modification of the mixing parameter in , the upper bound of the ratio varies in the range . Therefore, similar to the matrix , is a compression map for the general flavor triangle. After the flavor transition, the uncertainty of the flavor ratio at the source is decreased noticeably by .

For the quantitative illustration, we decompose the general flavor ratio at the source with the eigenvectors of , i.e.,

where satisfies the constraint from , namely, . The eigen-equations are listed as

with the eigenvalues expressed as

Here, denotes the trace of a matrix. The uncertainty of the flavor ratio at the source could be expressed as

After the flavor transition, the corresponding uncertainty of the flavor ratio at Earth is of the form

So the eigenvalues can serve as the indices of the compression ratio of the uncertainty of the flavor ratio at the source. We show the ranges of in Figure 3.

Figure 3: The eigenvalues of the matrix . The conventions of the mixing parameters are the same as those in Figure 2.

We can see that the flavor uncertainty index at Earth is around of the initial value. is several percent of the initial value. Particularly, if the eigenvalue is equal to 0, we will lose the information on the uncertainty of the flavor ratio expressed by . These observations still hold for the matrix with the modification of or . In fact, on the basis of the expression of shown in Eqs. (24)–(26), the nature of as a compression map is determined by the ranges of the mixing parameters in . This nature will not be changed with the improvement of the precision of the mixing parameters unless is constructed with the matrix completely different from .

3.2. Flavor Ratio at Earth with the Special Sources

Now we analyse the distribution of the flavor ratio at Earth with the special sources. We consider three typical sources, namely, the muon-damping source =, the neutron-decay source , and the pion-decay source . The flavor ratios at Earth with these sources are shown in Figures 4 and 5. From these figures, we can obtain the following observations:

(i)For every source, the impact of on the distribution of the flavor ratio at Earth is negligible because it is relatively small. Furthermore, the CP phases , in the matrix manifest themselves in the terms and , respectively. So, their impacts on the flavor ratio at Earth are undermined intensively by small 1-3 mixing angles in and . The dominant parameters in are , , , and . We note that a similar observation for the standard matrix has been obtained in the recent Ref. [52](ii)For the neutron-decay source and muon-damping source, , can enlarge the region of the flavor ratio at Earth noticeably. However, they cannot exempt the neutron-decay source from the constraint of the boundary of the 2015 IceCube data [53]. So, the new physics in is still not in favor of the neutron-decay as the unique source of the HAN(iii)For the pion-decay source, the impacts of , on the distribution of the flavor ratio at Earth are weakened by the approximate symmetry of the mixing matrix . Here, the symmetry denotes the property . As is known, on the basis of the strict symmetry and the standard transition matrix , the flavor ratio at Earth for the HAN from the pion-decay source is just [54, 55]. In our case that deviates from moderately, the flavor ratio at Earth also deviates from the democratic vector moderately(iv)From the light cyan regions in the figures, we can see that the observations in (i)-(iii) are not changed by the joint impacts of , in

Figure 4: The flavor ratio at Earth. (a) is for the neutron-decay source . (b) is for the muon-damping source . Except for the light cyan region, the conventions of the mixing parameters are the same as those in Figure 2. The light cyan region corresponds to the case that all the mixing parameters are modified in . The specific ranges for the modifications are listed as , , , and radian. The red region corresponds to the prediction of the standard matrix . The constraint on the flavor ratio is from the 2015 IceCube data [53].

Figure 5: The flavor ratio at Earth with the pion-decay source . The conventions of the mixing parameters are the same as those in Figure 4.

On the basis of the above observations, our setup of the matrix shows clear correlations between the distribution of the flavor ratio at Earth and the parameters of the mixing matrices. The dominant factors in are identified, namely, , . Their impacts on the flavor ratios can be discriminated. So the information on new physics can be read out from the distribution of the flavor ratio at Earth.

3.3. The Apparent P Violation Modulated by the CP Phase

In principle, the APV results from the mismatch between and . The mismatch is quantified by the modifications , . The correlations between the APV and the mixing parameters are analysed. We find that the APV is modulated by the Dirac phase, see Figures 6 and 7. From the figures, we obtain the following observations:

(i)The APV is modulated by the Dirac phase. This characteristic results from the expression of , see Eq. (28). In the case , the oscillation of is undermined. In the case , , we have when . Correspondingly, becomes maximal when is trivial. In the case , , the zero point of is around or (ii)The impact of the mass ordering of neutrinos on the APV is not negligible. The reason is that the range of at level with the normal mass ordering is wider than that with the inverted mass ordering, see the specific range of in Figures 6 and 7(iii)The oscillating amplitude of is determined by the magnitude of , , see Figure 8 for example. Compared with , the modification of the same order of can lead to the larger amplitude

Figure 6: The correlation between the apparent P violation (APV) and the CP phase. The mixing parameters in are in the ranges of the global fit data of the normal mass ordering [4]. The modified parameter in is fixed in every plot. The other parameters in are the same as those in .

Figure 7: The correlation between the apparent P violation (APV) and the CP phase with the inverted mass ordering. The conventions of the modified mixing parameters are the same as those in Figure 6.

Figure 8: The correlation between the apparent P violation (APV) with different amplitudes. The conventions of the mixing parameters are the same as those in Figure 6. The blue scatter points are for the case , the red points for the case , the green points for the case , and the black points for the case . (a) is for the case of the normal mass ordering. (b) is for the case of the inverted mass ordering.

On the basis of these observations, we find that the magnitude of is mainly dominated by the parameters , . Therefore, although is not important for the distribution of the flavor ration at Earth, we may still identify its impact on the APV when is negligible. So in the era of the precise detection the flavor composition of the HAN, we may obtain the information on through the secondary effect such as the APV.

3.4. Comment on Effects of Large Modifications in

Our observations on the flavor ratios of HAN at Earth and the APV are obtained in the case that differs from modestly. When large modification of mixing parameters in is considered, we could expect that the region of the flavor ratio distribution of HAN at Earth for the special source would be expanded noticeably. Furthermore, the correlation between APV and the Dirac CP phase would disappear under the large modification of . In special cases that only and are relatively large, the correlation could still be identified.

4. Conclusions

Inspired by the case of the solar neutrino oscillations, we proposed a generalized matrix to describe the flavor transition of the HAN. This matrix is constructed with in vacuum and at the source of the HAN. The mismatch between and leads to the nonsymmetry of which shows itself through the APV. In order to learn about the impacts of on the flavor ratio of the HAN at Earth and the APV, we studied the model-independent properties of with a vector-analysis method. On the basis of the unitarity of and , the generalized area theorem on the EFT is derived. The area of the EFT is determined by . With the setup that deviates from moderately, the impact of on the uncertainty of the flavor ratio at the source of the HAN is analysed. The range of and those of the eigenvalues of shows that the uncertainty at the source is decreased intensively by after the flavor transition of the HAN. The parameters in cannot change the nature of as the compression map of the general flavor triangle.

In the standard parametrization of and , the correlations between the parameters of and the distribution of the flavor ratio at Earth are shown. Compared with the results from the standard flavor transition matrix , we find that the dominant factors in for the flavor ratio are and . Furthermore, the impacts of and that of can be discriminated in the ternary plots of the flavor ratios at Earth. As a secondary effect of the new physics in , the APV is quantified by . The correlation between the and the mixing parameters in is obtained. We find that is modulated by the Dirac CP phase . So the information on may be obtained from the detection of the APV. In practice, is difficult to detect when the deviation of from is moderate. We propose a special case that may be identified through the flavor ratios at Earth with two distinguishable sources of the HAN, namely, . In general cases, this detection scenario may not work. However, the term is still a useful index to test the symmetry of the standard matrix in the future. Thus, our work may serve as a theoretical preparation for the era of the precise detection of the flavor composition of the HAN.

Data Availability

The data supporting this research paper are from previously reported studies, which have been cited. The processed data are freely available.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under grant nos. 12065007 and 11705113, the Guangxi Scientific Programme Foundation under grant no. Guike AD19110045, and the Research Foundation of Gunlin University of Technology under grant no. GUTQDJJ2018103.

References

Y. Fukuda, T. Hayakawa, E. Ichihara et al., “Evidence for oscillation of atmospheric neutrinos,” Physical Review Letters, vol. 81, no. 8, p. 1562, 1998.

Q. R. Ahmad, R. C. Allen, T. C. Andersen et al., “Direct evidence for neutrino flavor transformation from neutral-current interactions in the Sudbury Neutrino Observatory,” Physical Review Letters, vol. 89, article 011301, 2002.

F. P. An, J. Z. Bai, A. B. Balantekin et al., “Observation of electron-antineutrino disappearance at Daya Bay,” Physical Review Letters, vol. 108, no. 17, article 171803, 2012.

I. Esteban, M. C. González-García, M. Maltoni, T. Schwetz, and A. Zhou, “The fate of hints: updated global analysis of three-flavor neutrino oscillations,” Journal of High Energy Physics, vol. 2020, no. 9, article 178, 2020.

F. An, G. An, Q. An et al., “Neutrino physics with JUNO,” Journal of Physics G: Nuclear and Particle Physics, vol. 43, no. 3, article 030401, 2016.

K. Abe, K. Abe, H. Aihara et al., “Hyper-Kamiokande design report,” 2018, https://arxiv.org/abs/1805.04163.

B. Abi, R. Acciarri, M. A. Acero et al., “Volume I. Introduction to DUNE,” Journal of instrumentation, vol. 15, no. 8, article T08008, 2020.

M. G. Aartsen, R. Abbasi, Y. Abdou et al., “First observation of PeV-energy neutrinos with IceCube,” Physical review letters, vol. 111, article 021103, 2013.

M. E. Aartsen, M. Ackermann, J. Adams et al., “Observation of high-energy astrophysical neutrinos in three years of IceCube data,” Physical review letters, vol. 113, article 101101, 2014.

M. G. Aartsen, K. Abraham, M. Ackermann et al., “Evidence for astrophysical muon neutrinos from the northern sky with IceCube,” Physical review letters, vol. 115, article 081102, 2015.

R. Abbasi, M. Ackermann, J. Adams et al., “IceCube high-energy starting event sample: description and flux characterization with 7.5 years of data,” Physical Review D, vol. 104, article 022002, 2021.

J. Beringer, J. F. Arguin, R. M. Barnett et al., “Review of particle physics,” Physical Review D-Particles, Fields, Gravitation and Cosmology, vol. 86, article 010001, 2012.

C. A. Argüelles, T. Katori, and J. Salvado, “Effect of new physics in astrophysical neutrino flavor,” Physical Review Letters, vol. 115, no. 16, article 161303, 2015.

K. S. Babu, V. Brdar, A. de Gouvêa, and P. A. N. Machado, “Energy-dependent neutrino mixing parameters at oscillation experiments,” 2021, https://arxiv.org/abs/2108.11961.

J. F. Beacom, N. F. Bell, D. Hooper, S. Pakvasa, and T. J. Weiler, “Decay of high-energy astrophysical neutrinos,” Physical Review Letters, vol. 90, no. 18, article 181301, 2003.

D. Meloni and T. Ohlsson, “Neutrino flux ratios at neutrino telescopes: the role of uncertainties of neutrino mixing parameters and applications to neutrino decay,” Physical Review D, vol. 75, no. 12, article 125017, 2007.

P. Mehta and W. Winter, “Interplay of energy dependent astrophysical neutrino flavor ratios and new physics effects,” Journal of Cosmology and Astroparticle Physics, vol. 2011, no. 3, p. 41, 2011.

P. Baerwald, M. Bustamante, and W. Winter, “Neutrino decays over cosmological distances and the implications for neutrino telescopes,” Journal of Cosmology and Astroparticle Physics, vol. 2012, no. 10, p. 20, 2012.

G. Pagliaroli, A. Palladino, F. L. Villante, and F. Vissani, “Testing nonradiative neutrino decay scenarios with IceCube data,” Physical Review D, vol. 92, no. 11, article 113008, 2015.

M. Bustamante, J. F. Beacom, and K. Murase, “Testing decay of astrophysical neutrinos with incomplete information,” Physical Review D, vol. 95, no. 6, article 063013, 2017.

S. Choubey, S. Goswami, and D. Pramanik, “A study of invisible neutrino decay at DUNE and its effects on measurement,” Journal of High Energy Physics, vol. 2018, no. 2, article 55, 2018.

P. B. Denton and I. Tamborra, “Invisible neutrino decay could resolve IceCube’s track and cascade tension,” Physical Review Letters, vol. 121, no. 12, article 121802, 2018.

S. Antusch, C. Biggio, E. Fernandez-Martinez, M. B. Gavela, and J. Lopez-Pavon, “Unitarity of the leptonic mixing matrix,” ournal of High Energy Physics, vol. 2006, no. 10, article 84, 2006.

Z.-z. Xing and S. Zhou, “Implications of leptonic unitarity violation at neutrino telescopes,” Physics Letters B, vol. 666, no. 2, pp. 166–172, 2008.

X.-J. Xu, H.-J. He, and W. Rodejohann, “Constraining astrophysical neutrino flavor composition from leptonic unitarity,” Journal of Cosmology and Astroparticle Physics, vol. 2014, article 39, p. 12, 2014.

S. Parke and M. Ross-Lonergan, “Unitarity and the three flavor neutrino mixing matrix,” Physical Review D, vol. 93, no. 11, article 113009, 2016.

V. Brdar, J. Kopp, and X.-P. Wang, “Sterile neutrinos and flavor ratios in IceCube,” Journal of Cosmology and Astroparticle Physics, vol. 2017, article 1, p. 26, 2017.

J. Tang, Y. Zhang, and Y.-F. Li, “Probing direct and indirect unitarity violation in future accelerator neutrino facilities,” Physics Letters B, vol. 774, pp. 217–224, 2017.

C. A. Argüelles, K. Farrag, T. Katori, R. Khandelwal, S. Mandalia, and J. Salvado, “Sterile neutrinos in astrophysical neutrino flavor,” Journal of Cosmology and Astroparticle Physics, vol. 2020, article 15, p. 2, 2020.

S. A. R. Ellis, K. J. Kelly, and S. W. Li, “Current and future neutrino oscillation constraints on leptonic unitarity,” Journal of High Energy Physics, vol. 2020, article 68, 2020.

M. Kobayashi and C. S. Lim, “Pseudo Dirac scenario for neutrino oscillations,” Physical Review D, vol. 64, no. 1, article 013003, 2001.

J. F. Beacom, N. F. Bell, D. Hooper, J. G. Learned, S. Pakvasa, and T. J. Weiler, “Pseudo-Dirac neutrinos: a challenge for neutrino telescopes,” Physical Review Letters, vol. 92, no. 1, article 011101, 2004.

A. Esmaili, “Pseudo-Dirac neutrino scenario: cosmic neutrinos at neutrino telescopes,” hysical Review D, vol. 81, no. 1, article 013006, 2010.

A. Esmaili and Y. Farzan, “Implications of the pseudo-Dirac scenario for ultra high energy neutrinos from GRBs,” ournal of Cosmology and Astroparticle Physics, vol. 2012, article 12, p. 14, 2012.

A. S. Joshipura, S. Mohanty, and S. Pakvasa, “Pseudo-Dirac neutrinos via a mirror world and depletion of ultrahigh energy neutrinos,” Physical Review D, vol. 89, no. 3, article 033003, 2014.

V. Brdar and R. S. L. Hansen, “IceCube flavor ratios with identified astrophysical sources: towards improving new physics testability,” JCAP, vol. 2019, article 2, p. 23, 2019.

E. W. Kolb and M. S. Turner, “Supernova 1987A and the secret interactions of neutrinos,” Physical Review D, vol. 36, no. 10, pp. 2895–2900, 1987.

K. Ioka and K. Murase, “IceCube PeV–EeV neutrinos and secret interactions of neutrinos,” Progress of Theoretical and Experimental Physics, vol. 2014, no. 6, article 061E01, 2014.

K. C. Y. Ng and J. F. Beacom, “Cosmic neutrino cascades from secret neutrino interactions,” Physical Review D, vol. 90, no. 6, article 065035, 2014.

M. Bustamante, C. Rosenstrøm, S. Shalgar, and I. Tamborra, “Bounds on secret neutrino interactions from high-energy astrophysical neutrinos,” Physical Review D, vol. 101, article 123024, 2020.

M. Bustamante, J. F. Beacom, and W. Winter, “Theoretically palatable flavor combinations of astrophysical neutrinos,” Physical Review Letters, vol. 115, no. 16, article 161302, 2015.

L. Wolfenstein, “Neutrino oscillations in matter,” Physical Review D, vol. 17, no. 9, pp. 2369–2374, 1978.

S. P. Mikheyev and A. Y. Smirnov, “Resonance Amplification of Oscillations in Matter and Spectroscopyof Solar Neutrinos,” Soviet Journal of Nuclear Physics, vol. 42, p. 913, 1985.

V. D. Barger, K. Whisnant, S. Pakvasa, and R. J. N. Phillips, “Matter effects on three-neutrino oscillations,” Physical Review D, vol. 22, no. 11, pp. 2718–2726, 1980.

S. Toshev, “Resonant amplification of three-neutrino oscillations in matter,” Physics Letters B, vol. 185, no. 1-2, pp. 177–182, 1987.

T. Ohlsson, “Status of non-standard neutrino interactions,” Reports on Progress in Physics, vol. 76, no. 4, article 044201, 2013.

O. G. Miranda and H. Nunokawa, “Non standard neutrino interactions: current status and future prospects,” New Journal of Physics, vol. 17, no. 9, article 095002, 2015.

J. Liao and Y. Zhang, “Constraining nonstandard neutrino interactions at electron colliders,” Physical Review D, vol. 104, no. 3, article 035043, 2021.

C. Giunti and C. W. Kim, Fundamentals of Neutrino Physics and Astrophysics, Oxford university press, 2007.

C. Lunardini and A. Y. Smirnov, “The minimum width condition for neutrino conversion in matter,” Nuclear Physics B, vol. 583, no. 1-2, pp. 260–290, 2000.

L. Fu, C. M. Ho, and T. J. Weiler, “Aspects of the flavor triangle for cosmic neutrino propagation,” Physical Review D, vol. 91, no. 5, article 053001, 2015.

N. Song, S. W. Li, C. A. Argüelles, M. Bustamante, and A. C. Vincent, “The future of high-energy astrophysical neutrino flavor measurements,” Journal of Cosmology and Astroparticle Physics, vol. 2021, article 4, p. 54, 2021.

M. G. Aartsen, K. Abraham, M. Ackermann et al., “A combined maximum-likelihood analysis of the high-energy astrophysical neutrino flux measured with IceCube,” The Astrophysical Journal, vol. 809, no. 98, 2015.

J. G. Learned and S. Pakvasa, “Detecting oscillations at PeV energies,” Astroparticle Physics, vol. 3, no. 3, pp. 267–274, 1995.

Z.-Z. Xing and Z.-h. Zhao, “A review of flavor symmetry in neutrino physics,” Reports on Progress in Physics, vol. 79, no. 7, article 076201, 2016.